Strategic R&D in Cournot Markets - Fields Institute · 2013-08-27 · 4 Strategic R&D: Motivation...

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Strategic R&D: Motivation Dynamic R&D Control Extensions

Strategic R&D in Cournot Markets

Mike Ludkovski1 & Ronnie Sircar2

1UC Santa Barbara

2Princeton University

Fields InstituteAug 27, 2013

Ludkovski Strategic R&D in Cournot Markets

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Outline

1 Strategic R&D: MotivationCournot Games

2 Dynamic R&D ControlGame ModelSeparable ControlsCoupled Controls

3 Extensions


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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games

Resource and Commodities Markets

Long-term prices largely driven by production levels among several largeproducers

Lots of evidence of strategic behavior by participants

Noncooperative dynamic gameMany sources of uncertainty

◮ changing production costs◮ fluctuating demand◮ policy changes◮ technological advances

Fertile application area for stochastic games


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Role of R&D

Recently, attention has focused on exhaustibility: running out ofresources (peak oil)

Conversely, there has been a lot of technological breakthroughs:

Shale natural gas

Moore’s law for solar powerplants


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Role of R&D



Shale natural gas


Technological change is abrupt, ongoing, and uncertain

Aim to endogenize investment in R&D

Embed within a dynamic stochastic game framework


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Role of R&D



Shale natural gas


Technological change is abrupt, ongoing, and uncertain

Aim to endogenize investment in R&D

Embed within a dynamic stochastic game frameworkMotivation:

◮ Green (solar, biofuel) vs. fossil fuel energy production◮ R&D is key for switching to inexhaustible technologies◮ Game-theoretic effects can be significant


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Game Model

Cournot market: players control supply

Production levels qit ; production costs c i

t

Price is given by inverse demand curve P based on aggregate supply~q =

∑

i qi

Profit from production is qit · (P(~qt )− c i

t )

Each producer i looks at her total discounted revenue:

E

[∫ ∞

0e−rt {(P(~qt )− c i

t )qit − C(ai

t)}

dt]

.

ait is R&D effort

Look for closed-loop Markov Nash equilibrium


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Game Model

Cournot market: players control supply

Production levels qit ; production costs c i

t

Price is given by inverse demand curve P based on aggregate supply~q =

∑

i qi

Profit from production is qit · (P(~qt )− c i

t )

Each producer i looks at her total discounted revenue:

E

[∫ ∞

0e−rt {(P(~qt )− c i

t )qit − C(ai

t)}

dt]

.

ait is R&D effort

Look for closed-loop Markov Nash equilibrium

State : c it is stochastic, can be lowered through R&D

Everything else is deterministic


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Intuition

Recall the static Cournot duopoly with production costs c1 and c2

For simplicity, will focus on linear demand P(~q) = 1 −∑

i qi

The respective revenue is R1 := q1(1 − q1 − q2 − c1) andR2 := q2(1 − q1 − q2 − c2)

Interior eqm solution is qi,∗ = 1+c i−2c i

3 yielding revenue rate (qi,∗)2

Game value vi =(1+c i−2c i )2

9r


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Blockading

Production rate must benon-negative

If c i > 1+c i

2 , producer i isblockaded and does notproduce at all, q∗

i = 0. Inthat case have monopolywith q∗

i= (1 − c i)/2

Blockading when c i islarge (close to 1) relativeto c i . No blockading ifc i < 0.5

��

��

��

��

0.5

1

0.5

Player 1Blockaded

Blockaded

0

Player 2

1 c1

c2

R&D

Figure : Fixed-cost Cournot game.


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Existing Literature

Industrial Organization: optimizing R&D investment by a monopolistfacing uncertainty

Within exhaustible resource context: Kamien and Schwartz (1978),Lafforgue (2008), numerous papers addressing climate changemitigation.

But: Only single agent – no game effects

Game Theory: impact of technological change on strategic competition

Fudenberg & Tirole (1985), Weeds (2002)

But: One-shot games – focus on coordination/preemption, no dynamiceffects

Cournot Games: Hotelling (1931), Sircar et al. (2010–)

But: no R&D – production costs are fixed


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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls

R&D Control

Model technology as a discrete ladder: c(1) > c(2) > ... ≥ 0

If currently at n-th stage, a breakthrough moves the producer to n + 1-ststage of technology

c(n) = exp(−bn): efficiency improves proportionally by b%

c(n) = (1 − bn)+: absolute improvements in efficiency – eventually willreach “zero” costs


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R&D Control

Model technology as a discrete ladder: c(1) > c(2) > ... ≥ 0

If currently at n-th stage, a breakthrough moves the producer to n + 1-ststage of technology

c(n) = exp(−bn): efficiency improves proportionally by b%

c(n) = (1 − bn)+: absolute improvements in efficiency – eventually willreach “zero” costs

Expend effort at ⇒ breakthroughs occur at rate λat

N it : point process for the technology advances of player i. Given (ai

t ), (Nit )

is a Poisson process with controlled intensity λiait

Dynamic production costs are c it = c(N i

t )

R&D incurs costs C(at) per unit time; convex + increasing

For convenience assume finite number of stages Ni for player i


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Dynamic R&D

Only uncertainty is from (N it ). Between R&D advances the game is

deterministic. Can be viewed as a sequence of coupled static Cournotgames

There is dynamic interaction between R&D and production. Players maybe blockaded, and may also choose to expend no effort (making somestates (c1, c2) absorbing)

Continuous-time strategies: qit , a

it

Strategies assumed to be in feedback form for (N1t ,N

2t )


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Finding Nash Equilibrium

Given initial technology stages (n1, n2), game values are denoted byvi(n1, n2)

τ i is the time of first R&D success by player i – controlled by effort (ait )

Given (qi , ai ), vi ’s satisfy the recursions

v1(n1, n2) = E

[

∫ τ1∧τ

2

0e−rs{q1

s (P(~qs)− c1(n1))− C(a1s)} ds

+ e−rτ 1∧τ2[1{τ 1<τ 2} · v1(n1 + 1, n2) + 1{τ 1>τ 2} · v1(n1, n2 + 1)]

]

By the piecewise deterministic property, under every Markov Nashequilibrium, qi

t ≡ qi , ait ≡ ai are constant for t ∈ [0, τ1 ∧ τ2]

So τ1 ∧ τ2 ∼ Exp(λ1a1 + λ2a2)


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Duopoly Game Values

Using properties of Poisson arrival times, Nash equilibria arecharacterized by

v1(n1, n2) = supq,a

q(1− q − q2,∗ − c1(n1))− C(a) + λ1av1(n1 + 1,n2) + λ2a2,∗v1(n1, n2 + 1)λ1a + λ2a2,∗ + r

Similar equation for v2(n1, n2)

q1,∗ is obtained directly as 1+c1(n1)−2c2(n2)3

Differentiating wrt a’s, obtain a system of two nonlinear equations ina1,∗, a2,∗ characterizing the Nash equilibrium


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Recursive Static Games

v1(n1, n2 + 1)

yλ2a2

v1(n1, n2)λ1a1

←−−−−− v1(n1 + 1, n2)

Can solve recursively on a lattice

Boundary condition is vi(N1,N2) =(1+c1(N1)−2c2(N2))

2

9r ; also when n1 = N1,no further R&D is possible for P1 (1-dim optim by P2)

a1,∗ depends on v1(n1 + 1, n2)− v1(n1, n2) > 0 andv1(n1, n2 + 1)− v1(n1, n2) < 0

C(a) = a2/2 + κa:◮ Have a system of two coupled quadratic equations for ai,∗

◮ if κ > 0 then R&D may be unprofitable, so a∗ = 0 is possible◮ Analytic expressions to determine whether R&D is zero


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Unilateral R&D

0.00

0.05

0.10

0.15

0.20

0.25

P1 Costs c_1

R&D

Effo

rt

0.75 0.49 0.38 0.30 0.23 0.17 0.11 0.06 0.02

c_2 = 0.5c_2 = 0.25c_2 = 0

Figure : Effort Curves for Unilateral R&D in a Cournot Duopoly. Quadratic effort costC(a) = a2/2 + 0.2a with λ = 5, r = 0.1. Here c1(n) = 0.75− 1.5

√n (q1(n) is linear)


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Unilateral R&D

0.00

0.02

0.04

0.06

0.08

0.10

0.12

P1 Costs c_1

R&

D E

ffort

1.00 0.88 0.76 0.64 0.52 0.40 0.28 0.16 0.04

P2

Blo

ckad

ed

P1

Blo

ckad

ed

Levels of R&D over time may have different shapes

Affected by: expectation of future profits (less future gains as get closeto c1 = 0), shape of the cost curve n 7→ c1(n) (marginal efficiency ofR&D), current revenue levels

If initial c1 is too high relative to c2, become too discouraged and donothing (never enter the market)

Monopoly is more conducive to R&D than duopoly


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Bilateral R&D

R&D Effort

Costs P1

Cos

ts P

2

0.78 0.37 0.17 0.08

0.78

0.37

0.17

0.08

Production

Costs P1

0.78 0.37 0.17 0.08

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Figure : Right panel shows the effort a1(n1, n2) and left panel the production rateq1(n1, n2). Quadratic costs C(a) = a2/2 + 0.2a with λ = 5, r = 0.1. c i(n) = e−n/8


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Bilateral R&D

R&D effort levels are asymmetric: put most effort when slightly ahead ofcompetitor

Therefore, player with lower costs tends to extend her advantage("mean-aversion")

Expectations of future profits can spur R&D even if currently blockadedout of the market

Zero R&D can happen even without blockading

Conversely, if c i is very low compared to competitor, may becomecomplacent and stop R&D


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Bilateral R&D

R&D effort levels are asymmetric: put most effort when slightly ahead ofcompetitor

Therefore, player with lower costs tends to extend her advantage("mean-aversion")

Expectations of future profits can spur R&D even if currently blockadedout of the market

Zero R&D can happen even without blockading

Conversely, if c i is very low compared to competitor, may becomecomplacent and stop R&D

Outside input (subsidies) can spur endogenous advances both for veryinefficient technologies and for efficient monopolies

Competition is dynamically unstable (tends to collapse into a monopoly)


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Sample Path of (N1t ,N

2t )

0 10 20 30 40 50

510

1520

25

Time

Tech

nolo

gy S

tage

P1P2

Figure : Sample path of (N1t ,N

2t ). Costs are c i(n) = e−n/8. Quadratic effort curve

C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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Distribution of (N1t ,N

2t )

t = 2

5 10 15 20 25

510

1520

25

P1 Technology Stage

P2

Tech

nolo

gy S

tage

Figure : Distribution of (N1t ,N


C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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2t )

t = 4

5 10 15 20 25

510

1520

25

P1 Technology Stage

P2

Tech

nolo

gy S

tage



C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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2t )

t = 8

5 10 15 20 25

510

1520

25

P1 Technology Stage

P2

Tech

nolo

gy S

tage



C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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2t )

t = 15

5 10 15 20 25

510

1520

25

P1 Technology Stage

P2

Tech

nolo

gy S

tage



C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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2t )

t = 25

5 10 15 20 25

510

1520

25

P1 Technology Stage

P2

Tech

nolo

gy S

tage



C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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2t )

t = 40

5 10 15 20 25

510

1520

25

P1 Technology Stage

P2

Tech

nolo

gy S

tage



C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.


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Impact of Uncertainty

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

Game value

P1 Costs c^1

v_1

(c^1

,c^2

)

c^1(n) = 1−n/10c^1(n) = 1−n/20c^1(n) = 1−n/50

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

R&D Effort

P1 Costs c^1

a^*_

t

Figure : Comparison of game values v1(·, c2) and effort levels a1(·, c2). Lineartechnology progress c1(n) = 1− n/M, λ = 0.4M with c2 = 0.7 and κ = 0.2.

Can study effect of uncertainty by linearly scaling the R&D ladder c(n)and rate of progress λ

Take c(n) = f (n/M), λ = λM where c 7→ f (c) is cont R&D curve on [0, 1]As M → ∞, R&D success becomes deterministic: dct = λat dtImpact is ambiguous: more uncertainty can spur/deter R&D investment!


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R&D Complementary to Production

Firm has fixed labor supply L. Allocate L between production and R&D:ai

t + qit = Li

Sharpens the trade-off between immediate revenue and future higherprofits

Cost of R&D is now implicit (quadratic if assume linear demand P(~q))

Will tend to decrease R&D over time

May be optimal to voluntarily lower/suspend production to advancetechnology (e.g. to lock-in monopoly)

May allow a high-cost competitor to operate by temporarily focusing onR&D (i.e. strategic non-blockading)


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Extensions: Exhaustible Resources

When considering competition between old and new energy (fossil fuelsvs. renewables), exhaustible reserves play a crucial role

Xt – level of reserves at date t ; dXt = −qtdt lowered through production

Oil industry (P1): low production costs c1, but also marginal cost ofexhaustibility

Renewables industry (P2): high current production costs c2(0); potentialfor R&D

P1 chooses (q1t ); P2 chooses (q2

t ) and (a2t ). State is (x , n)

Leads to a system of nonlinear ODEs in x , coupled through n

Can allow P1 to also explore for new reserves (L. & Sircar 2012)


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Extensions: Switching Technologies

Consider two integrated producers who can each use either cheap fossilfuels, or expensive backstops (oil sands)

Resources allocated between production and R&D (advancing backstoptechnology)

Uncertainty in advances will spur earlier R&D investments as marginalvalue of cheap reserves rises

Related to the model of Harris, Howison and Sircar (2010)


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Conclusion

Ongoing project: stochastic framework for natural resource oligopolies

Effect of exhaustibility

Potential of R&D

Strategic interactions (blockading, uncertainty, policy impact) lead tonumerous non-trivial phenomena


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Conclusion

Ongoing project: stochastic framework for natural resource oligopolies

Effect of exhaustibility

Potential of R&D

Strategic interactions (blockading, uncertainty, policy impact) lead tonumerous non-trivial phenomena

THANK YOU!


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References

C. Harris, S. Howison, and R. SircarGames with exhaustible resourcesSIAM J. Applied Mathematics 70 (2010), 2556-2581.

G. LafforgueStochastic technical change, exhaustible resource and optimal sustainable growthResource and Energy Economics 30 (2008), no. 4, 540–554.

J. WeedsStrategic delay in a real options model of R&D competitionReview of Economic Studies, 69 (2002), 729–747.

M. Ludkovski and R. SircarExploration and Exhaustibility in Dynamic Cournot GamesEuropean Journal of Applied Mathematics, 23 (2012), no. 3, 343–372.


Strategic R&D in Cournot Markets - Fields Institute · 2013-08-27 · 4 Strategic R&D: Motivation...

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